In this paper, we analyze the supercloseness result of nonsymmetric interior penalty Galerkin (NIPG) method on Shishkin mesh for a singularly perturbed convection diffusion problem. According to the characteristics of the solution and the scheme, a new analysis is proposed. More specifically, Gau{\ss} Lobatto interpolation and Gau{\ss} Radau interpolation are introduced inside and outside the layer, respectively. By selecting special penalty parameters at different mesh points, we further establish supercloseness of almost k + 1 order under the energy norm. Here k is the order of piecewise polynomials. Then, a simple post processing operator is constructed. In particular, a new analysis is proposed for the stability analysis of this operator. On the basis of that, we prove that the corresponding post-processing can make the numerical solution achieve higher accuracy. Finally, superconvergence can be derived under the discrete energy norm. These theoretical conclusions can be verified numerically.
翻译:在本文中, 我们分析在希什金网格中, 使用非对称内部惩罚Galerkin( NIPG) 方法的超近距离结果, 是一个特别受扰动的对流扩散问题。 根据解决方案和计划的特点, 提议进行新的分析。 更具体地说, 分别在该层内外引入了Lobatto 内插和 Gauss} Radau 内插。 通过在不同网格点选择特殊惩罚参数, 我们进一步在能源规范下建立几乎 k+ 1 命令的超近距离。 这里 k 是一个小圆形的多面体顺序 。 然后, 构建了一个简单的后端处理操作器。 特别是, 为该操作器的稳定性分析建议了新的分析 。 在此基础上, 我们证明相应的后处理可以提高数字解决方案的精确度。 最后, 可以在离散能源规范下生成超级相容值。 这些理论结论可以被数校准 。